Abstract
Let be the vector spaces and V_i^* be their dual spaces for 1≤i≤r . In this paper, (V_1×V_2×…×V_r )^*≅V_1^*×V_2^*×…×V_r^* is examined. Furthermore, the dual map F^* and the adjoint map F^' of the r-linear map F:V_1×V_2×…×V_r→W are defined and for their matrices, the equality [F^* ]=[F^' ]=[F ̅ ]^T is found. Analog, the dual map F^* of the r-variable vector valued linear map F:V_1×V_2×…×V_r→W_1×W_2×…×W_r is defined and for their matrices, the equality [F^* ]=[F ̅ ]^T is true, for the vector spaces V_i and W_i .
Keywords
Mathematics
Multilinear map
Dual space
Vector map
Dual (grammatical number)
Combinatorics
Linear map
Vector space
Discrete mathematics
Geometry
Pure mathematics
Computer science
Artificial intelligence
Subject Areas
graph theory and CDMA systems
·Electrical and Electronic Engineering ·Physical Sciences
Advanced Topics in Algebra
·Algebra and Number Theory ·Physical Sciences
Matrix Theory and Algorithms
·Computational Theory and Mathematics ·Physical Sciences
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